Description
A Kombinatorika szemináriumon október 16-án 2:15-kor a Rényi Intézet Nagytermében
For a graph $F$, an $r$-uniform hypergraph ($r$-graph for short) $cH$ is a Berge-$F$ if there is a bijection $\phi:E(F)\rightarrow E(cH)$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-graphs, an $r$-graph is $\mathcal{F}$-free if it does not contain any member in $\mathcal{F}$ as a subhypergraph. The Tur\'{a}n number of $\mathcal{F}$ is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-graph on $n$ vertices.
Kang, Ni, and Shan [Discrete Math. 345 (2022) 112901] determined the exact value of the Tur\'{a}n number of Berge-$M_{s+1}$ for all $n$ when $r\leq s-1$ or $r\geq 2s+2$, where $M_{s+1}$ denotes a matching of size $s+1$.
In this paper, we settle the remaining case $s\le r\le 2s+1$. Moreover, we establish several exact and general results on the Tur\'{a}n numbers of Berge matchings together with a single $r$-graph, as well as of Berge matchings together with Berge bipartite graphs. Finally, we generalize the results on Tur\'{a}n problems for Berge hypergraphs proposed by Gerbner, Methuku, and Palmer [Eur. J. Comb. 86 (2020) 103082].
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